On linear independence of Dirichlet $L$ values

TL;DR

This paper investigates the linear independence of Dirichlet L-values at fixed integers across families of characters with different moduli, extending previous fixed-modulus results and addressing new technical challenges in number field arithmetic.

Contribution

It extends prior work on linear independence of Dirichlet L-values to families of characters with coprime moduli, involving detailed analysis of compositum fields.

Findings

Extended linear independence results to multiple moduli

Developed methods for arithmetic of compositum number fields

Addressed technical challenges in field interactions

Abstract

The study of linear independence of $L(k, \chi)$ for a fixed integer $k>1$ and varying $\chi$ depends critically on the parity of $k$ vis-\`a-vis $\chi$. This has been investigated by a number of authors for Dirichlet characters $\chi$ of a fixed modulus and having the same parity as $k$.The focal point of this article is to extend this investigation to families of Dirichlet characters modulo distinct pairwise co-prime natural numbers. The interplay between the resulting ambient number fields brings in new technical issues and complications hitherto absent in the context of a fixed modulus (consequently a single number field lurking in the background). This entails a very careful and hands-on dealing with the arithmetic of compositum of number fields which we undertake in this work. Our results extend earlier works of the first author with Murty-Rath as well as works of Okada, Murty-Saradha and Hamahata.